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Set of quantities in probability theory
have identical cumulants as well, and vice versa. The first cumulant is the mean, the second cumulant is the variance, and the third cumulant is the same
Cumulant
Family of probability distributions
obtain the cumulant function for different cases of the Tweedie models. A cumulant generating function (CGF) may then be obtained from the cumulant function
Tweedie_distribution
Kurtosis of the order parameter in statistical physics
The Binder parameter or Binder cumulant in statistical physics, also known as the fourth-order cumulant U L = 1 − ⟨ s 4 ⟩ L 3 ⟨ s 2 ⟩ L 2 {\displaystyle
Binder_parameter
Measure of the asymmetry of random variables
the third cumulant κ3 to the 1.5th power of the second cumulant κ2. This is analogous to the definition of kurtosis as the fourth cumulant normalized
Skewness
Probability distribution modeling a coin toss which need not be fair
_{6}&=\mu _{2}(1-5\mu _{2}(1-\mu _{2})).\end{aligned}}} The first six cumulants are κ 1 = p , κ 2 = μ 2 , κ 3 = μ 3 , κ 4 = μ 2 ( 1 − 6 μ 2 ) , κ 5 =
Bernoulli_distribution
Unbiased estimator of a cumulant
a k-statistic is a statistic constructed from a sample to estimate a cumulant of the underlying population distribution. For a sample of independent
K-statistic
the calculation of cumulants, for what two distinguishable ways exist. For one thing an image can be calculated via auto-cumulants that by definition
Super-resolution optical fluctuation imaging
Super-resolution_optical_fluctuation_imaging
Probability distribution
{\displaystyle \operatorname {Li} _{-n}(1-p)} is the polylogarithm function. The cumulant generating function of the geometric distribution defined over N 0 {\displaystyle
Geometric_distribution
Probability distribution
{\displaystyle \alpha ^{\overline {r}}} the rising factorial. The r-th cumulant is given by: κ r = θ r α ( r − 1 ) ! = θ r α Γ ( r ) {\displaystyle \kappa
Gamma_distribution
Family of probability distributions related to the normal distribution
{\displaystyle K{\left(u\mid \eta \right)}=A(\eta +u)-A(\eta )\,,} is the cumulant generating function of the sufficient statistic. Exponential families have
Exponential_family
Fourth standardized moment in statistics
is κ, which is fine as long as it is clear that it does not refer to a cumulant. Other choices include γ2, to be similar to the notation for skewness,
Kurtosis
Power series derived from a discrete probability distribution
include the moment-generating function, the characteristic function and the cumulant generating function. The probability generating function is also equivalent
Probability generating function
Probability_generating_function
Moment of a random variable minus its mean
n ≥ 4 is the n-th cumulant κn(X). For n = 1, the n-th cumulant is just the expected value; for n = either 2 or 3, the n-th cumulant is just the n-th central
Central_moment
Technique for determining size distribution of particles
from an autocorrelation function. One of the most common methods is the cumulant method, from which in addition to the sum of the exponentials above, more
Dynamic_light_scattering
Measure of the shape of a probability distribution function
the next section, excess kurtosis is the fourth cumulant divided by the square of the second cumulant.) If a distribution has heavy tails, the kurtosis
Moment_(statistics)
Description of continuous random distribution
deviation skewness kurtosis L-moment moment generating function (mgf) characteristic function probability generating function (pgf) cumulant combinant
Probability_density_function
Indian theoretical chemist
Debashis Mukherjee (born 17 December 1946) is an Indian theoretical chemist, well known for his research in the fields of molecular many body theory, theoretical
Debashis_Mukherjee
Higher-order frequency analysis
stationary process (the signal) z ( t ) {\displaystyle z(t)} and its multi-time cumulant. f ( τ 1 , ⋯ , τ n − 1 ) = C n ( z ( t ) , z ( t + τ 1 ) , ⋯ , z ( t +
Polyspectra
Probability distribution
free probability theory, the role of cumulants is occupied by "free cumulants", whose relation to ordinary cumulants is simply that the role of the set
Wigner semicircle distribution
Wigner_semicircle_distribution
Family of continuous probability distributions
the cumulant generating functions of the Gaussian and inverse Gaussian distributions are inverse of each other (i.e., the graphs of the two cumulant generating
Inverse_Gaussian_distribution
Fourier transform of the probability density function
characteristic function is a cumulant generating function, which is useful for finding cumulants; some instead define the cumulant generating function as the
Characteristic function (probability theory)
Characteristic_function_(probability_theory)
Infinite sum approximating a probability distribution in terms of its cumulants
real line ( − ∞ , ∞ ) {\displaystyle (-\infty ,\infty )} in terms of its cumulants. The series are the same; but, the arrangement of terms (and thus the
Edgeworth_series
mathematical statistics, the law of total cumulance is a generalization to cumulants of the law of total probability, the law of total expectation, and the
Law_of_total_cumulance
Probability distribution and special case of gamma distribution
{k}{2}}\right)}}{\Gamma {\left({\frac {k}{2}}\right)}}}.\end{aligned}}} The cumulants are readily obtained by a power series expansion of the logarithm of the
Chi-squared_distribution
Concept in probability theory and statistics
therefore be deduced from it by inverse Fourier transform. Cumulant-generating function The cumulant-generating function is defined as the logarithm of the
Moment_generating_function
Probability that random variable X is less than or equal to x
deviation skewness kurtosis L-moment moment generating function (mgf) characteristic function probability generating function (pgf) cumulant combinant
Cumulative distribution function
Cumulative_distribution_function
Class of probability distributions
{\theta }}+\mathbf {t} )-A({\boldsymbol {\theta }})\ {\Big )}\,.} The cumulant generating function is by definition the logarithm of the MGF, so it is
Natural_exponential_family
Average value of a random variable
deviation skewness kurtosis L-moment moment generating function (mgf) characteristic function probability generating function (pgf) cumulant combinant
Expected_value
Polynomials in combinatorial mathematics
whose first n cumulants are κ1, ..., κn. In other words, the nth moment is the nth complete Bell polynomial evaluated at the first n cumulants. Likewise,
Bell_polynomials
observation. The cumulants of the sum of the grouped variable and the uniform variable are the sums of the cumulants. As odd cumulants of a uniform distribution
Sheppard's_correction
Measure of the deviation of position over time
natural log of the characteristic function, a new function is produced, the cumulant generating function, ln ( G ( k ) ) = ∑ m = 1 ∞ ( i k ) m m ! κ m , {\displaystyle
Mean_squared_displacement
Mathematical theory
Important features in common with the cumulants are: the combinants share the additivity property of the cumulants; for infinite divisibility (probability)
Combinant
Set of probability distributions
{\theta }}} has the same dimension as X {\displaystyle \mathbf {X} } . The cumulant-generating function of Y ∼ E D ( μ , σ 2 ) {\displaystyle Y\sim \mathrm
Exponential_dispersion_model
Computational statistics technique
M_{X}(t)|_{t=\theta }} . It is easy to derive the cumulant-generation function of the proposal and therefore the proposal's cumulants. ψ θ ( η ) = log ( E θ exp (
Rejection_sampling
mechanics, an Ursell function or connected correlation function, is a cumulant of a random variable. It can often be obtained by summing over connected
Ursell_function
Type of polynomial sequence
termed the cumulants of the polynomial sequence. It can be shown that the whole polynomial sequence of binomial type is determined by its cumulants, in a way
Binomial_type
Statistical measure of how far values spread from their average
(X)=\operatorname {Cov} (X,X).} The variance is also equivalent to the second cumulant of a probability distribution that generates X {\displaystyle X} . The
Variance
Infinite series used to approximate quantiles of probability distributions
approximate the quantiles of a probability distribution based on its cumulants. It is named after E. A. Cornish and R. A. Fisher, who first described
Cornish–Fisher_expansion
Discrete probability distribution
When λ is a positive integer, the modes are λ and λ − 1. All of the cumulants of the Poisson distribution are equal to the expected value λ. The n-th
Poisson_distribution
Discrete-variable probability distribution
deviation skewness kurtosis L-moment moment generating function (mgf) characteristic function probability generating function (pgf) cumulant combinant
Probability_mass_function
Order of Sufism
might have suggested. Rather, their joint effect is to impart to Sufism a cumulant body of tradition, rather than individual and isolated experiences. In
Tariqa
Compound Poisson-family discrete probability distribution
{\displaystyle M(t)=G_{Y}(e^{t})=\exp(\lambda (e^{\phi (e^{t}-1)}-1))} The cumulant generating function is the logarithm of the moment generating function
Neyman_Type_A_distribution
Noncentral generalization of the chi-squared distribution
{\displaystyle \mu _{4}=12(k+2\lambda )^{2}+48(k+4\lambda )\,} The nth cumulant is κ n = 2 n − 1 ( n − 1 ) ! ( k + n λ ) . {\displaystyle \kappa _{n}=2^{n-1}(n-1)
Noncentral chi-squared distribution
Noncentral_chi-squared_distribution
Mathematical function for the probability a given outcome occurs in an experiment
deviation skewness kurtosis L-moment moment generating function (mgf) characteristic function probability generating function (pgf) cumulant combinant
Probability_distribution
Measure of variation in statistics
An inequality on location and scale parameters Coefficient of variation Cumulant Deviation (statistics) Distance correlation Distance standard deviation
Standard_deviation
Uniform distribution on an interval
} For n ≥ 2 , {\displaystyle n\geq 2,} the n {\displaystyle n} -th cumulant of the continuous uniform distribution on the interval [ − 1 2 , 1 2
Continuous uniform distribution
Continuous_uniform_distribution
Probability distribution
1;\lambda )}}.} Many important summary statistics, such as moments and cumulants, of the CMP distribution can be expressed in terms of the normalizing
Conway–Maxwell–Poisson distribution
Conway–Maxwell–Poisson_distribution
Smooth approximation to the maximum function
differentiable). It is encountered in machine learning, for example, as the cumulant of the multinomial/binomial family. In tropical analysis, this is the sum
LogSumExp
Probability distribution
power series define the cumulants, but because this is a quadratic polynomial in t {\displaystyle t} , only the first two cumulants are nonzero, namely
Normal_distribution
univariate and multivariate distributions is through the use of cumulants and joint cumulants. In time series analysis, the extension of these is to polyspectra
Higher-order_statistics
Statistical probability Distribution for discrete event counts
{\displaystyle M(t)=G(e^{t})=\exp(a_{1}(e^{t}-1)+a_{2}(e^{2t}-1))} The cumulant generating function is the logarithm of the moment generating function
Hermite_distribution
Topics referred to by the same term
consumer goods companies Cumulant generating function expression for defining statistical mean, variance, and higher-order cumulants. Cuyahoga County Airport
CGF
Product of numbers from 1 to n
p. 215. Daley, D. J.; Vere-Jones, D. (1988). "5.2: Factorial moments, cumulants, and generating function relations for discrete distributions". An Introduction
Factorial
} , the population correlation ρ {\textstyle \rho } , the population cumulants κ r {\textstyle \kappa _{r}} , A tilde (~) denotes "has the probability
Notation in probability and statistics
Notation_in_probability_and_statistics
Statistic used for nonlinear interactions
transform of C3(t1, t2) (third-order cumulant-generating function). The Fourier transform of the second-order cumulant, i.e., the autocorrelation function
Bispectrum
Mathematical result
{1}{k}}\sum _{i}Q_{i}^{2}} around 1. This requires upper-bounding the cumulant generating function (CGF). Moment bounds (Achlioptas, 2003, Section 6)—For
Johnson–Lindenstrauss_lemma
Statistical function that defines the quantiles of a probability distribution
deviation skewness kurtosis L-moment moment generating function (mgf) characteristic function probability generating function (pgf) cumulant combinant
Quantile_function
Artificial intelligence VTuber
January 2026). "«C'est ça, le futur du streaming?»: sur Twitch, la chaîne cumulant le plus d'abonnés payants est animée par une IA" ["Is this the future of
Neuro-sama
Theorem in probability theory
Fyodorov-Bouchaud formula. For non-Gaussian random variables, the moment-cumulants formula replaces the Wick's probability formula. If ( X 1 , … X n ) {\displaystyle
Isserlis's_theorem
Measure of phase coupling in a signal
second-order cumulant, i.e., the autocorrelation function, is the traditional power spectrum. The Fourier transform of C3(t1,t2) (third-order cumulant) is called
Bicoherence
Expectation or average of the falling factorial of a random variable
numbers of the second kind. Factorial moment measure Moment (mathematics) Cumulant Factorial moment generating function The Pochhammer symbol (x)r is used
Factorial_moment
Discrete probability distribution
_{1}&=\Delta /(2\mu )^{3/2},\\[4pt]\gamma _{2}&=1/2.\end{aligned}}} The cumulant-generating function is given by: K ( t ; μ 1 , μ 2 ) = d e f ln (
Skellam_distribution
Statistic for rank correlation
ISSN 0003-1305. Valz, Paul D.; McLeod, A. Ian; Thompson, Mary E. (February 1995). "Cumulant Generating Function and Tail Probability Approximations for Kendall's Score
Kendall rank correlation coefficient
Kendall_rank_correlation_coefficient
Regression models accounting for possible errors in independent variables
moments — the GMM estimator based on the third- (or higher-) order joint cumulants of observable variables. The slope coefficient can be estimated from β
Errors-in-variables_model
Name for several different families of probability distributions
first cumulant, κ 1 {\displaystyle \kappa _{1}} , is the mean and the second, κ 2 {\displaystyle \kappa _{2}} , is the variance. The third cumulant, κ 3
Generalized logistic distribution
Generalized_logistic_distribution
\Psi _{Q}^{*}} is the rate function, i.e. the convex conjugate of the cumulant-generating function, of Q {\displaystyle Q} , and μ 1 ′ ( P ) {\displaystyle
Kullback's_inequality
Rational mathematical function indexed by integer partitions
doi:10.1090/noti2474. ISSN 0002-9920. Collins, Benoît (2003), "Moments and cumulants of polynomial random variables on unitary groups, the Itzykson-Zuber integral
Weingarten_function
Probability distribution
for any even central moment can be found by first obtaining the even cumulants κ 2 n = 2 2 n − 1 ( 2 2 n − 1 ) B 2 n n ( 3 2 n − 1 ) , {\displaystyle
Cantor_distribution
Characteristic of conic sections
Classification of discrete distributions by variance-to-mean ratio; see cumulants of some discrete probability distributions for details. Classification
Eccentricity_(mathematics)
Concept in combinatorial mathematics
role in defining free cumulants in free probability theory that is played by the lattice of all partitions in defining joint cumulants in classical probability
Noncrossing_partition
Probability distribution
doi:10.1016/j.aam.2016.08.001. Perrault, P. (2024). "A New Bound on the Cumulant Generating Function of Dirichlet Processes". arXiv:2409.18621 [math.PR]
Dirichlet_distribution
Concept in information theory
is the large deviations rate function, i.e. the convex conjugate of the cumulant-generating function, of Q, and μ 1 ′ ( P ) {\displaystyle \mu '_{1}(P)}
Inequalities in information theory
Inequalities_in_information_theory
American physicist
physicist and professor of physics at Brown University. He has applied cumulant expansion methods to atmospheric and oceanic dynamics, bridging quantum
Brad_Marston
correlation, regression Thiele, Thorvald N. Danish 1838 1910 Introduced cumulants and the term "likelihood". Introduced a Kalman filter in time-series Peirce
Founders_of_statistics
Curve from a cone intersecting a plane
negative binomial distributions as hyperbolic. This is elaborated at cumulants of some discrete probability distributions. Confocal conic sections Circumconic
Conic_section
Probability distribution
m_{k+1}=rPm_{k}+(P^{2}+P){dm_{k} \over dP},\quad P:=(1-p)/p,\quad m_{0}=1.} For the cumulants κ k + 1 = ( Q − 1 ) Q d κ k d Q , Q := 1 / p , κ 1 = r ( Q − 1 ) . {\displaystyle
Negative binomial distribution
Negative_binomial_distribution
Branch of probability theory
\lambda (\theta )=\ln \operatorname {E} [\exp(\theta X)]} is called the cumulant generating function (CGF) and E {\displaystyle \operatorname {E} } denotes
Large_deviations_theory
Variable used for specification
is possible to use the sequence of moments (mean, mean square, ...) or cumulants (mean, variance, ...) as parameters for a probability distribution: see
Parameter
Bound on probability of a random variable being far from its mean
\left(e^{tX}\right),\qquad t>0.} Let K ( t ) {\displaystyle K(t)} be the cumulant generating function, K ( t ) = log ( E ( e t x ) ) . {\displaystyle
Chebyshev's_inequality
Exponentially decreasing bounds on tail distributions of random variables
equivalent to the Legendre–Fenchel transform or convex conjugate of the cumulant generating function K = log M {\displaystyle K=\log M} , defined as:
Chernoff_bound
Danish astronomer
contributions to the statistical study of random time series and introduced the cumulants and likelihood functions, and was considered to be one of the greatest
Thorvald_N._Thiele
partition Young's lattice Bell number Bell polynomials Dobinski's formula Cumulant Data clustering Equivalence relation Exact cover Knuth's Algorithm X Dancing
List_of_partition_topics
x=0} . The first cumulant κ 1 {\displaystyle \kappa _{1}} is equal to λ − r {\displaystyle \lambda -r} and all subsequent cumulants κ n , n ≥ 2 {\displaystyle
Displaced Poisson distribution
Displaced_Poisson_distribution
Mathematical theory on random variables
before in Wigner's semi-circle law in the random matrix context. The free cumulant functional (introduced by Roland Speicher) plays a major role in the theory
Free_probability
Method of utilizing water in magnetic resonance imaging
the presence of 2 water pools in slow or intermediate exchange and the cumulant-expansion (also called Kurtosis) model, which does not necessarily require
Diffusion-weighted magnetic resonance imaging
Diffusion-weighted_magnetic_resonance_imaging
Type of statistical analysis
histogram (PCH), fluorescence intensity distribution analysis (FIDA), and Cumulant Analysis. and Spatial Intensity Distribution Analysis. Combination of multiple
Fluorescence correlation spectroscopy
Fluorescence_correlation_spectroscopy
Rational number sequence
(n)={\frac {(-1)^{{\frac {n}{2}}-1}B_{n}(2\pi )^{n}}{2(n!)}}.} The nth cumulant of the uniform probability distribution on the interval [−1, 0] is Bn/n
Bernoulli_number
Kind of numerical parameter of a parametric family of probability distributions
Ekawati, Dian; Warsono; Kurniasari, Dian (December 2014). "On the Moments, Cumulants, and Characteristic Function of the Log-Logistic Distribution" (PDF).
Shape_parameter
Theorem In probability theory and statistics
f ( x ) {\displaystyle S=\sum _{x\in {N}}a_{n}f(x)} In this case the cumulants κ i {\displaystyle \kappa _{i}} of S {\displaystyle S} equal κ i = λ a
Campbell's theorem (probability)
Campbell's_theorem_(probability)
Normalized central moments
alternative definitions exist, which are based on the third and fourth cumulant respectively. Another scale invariant, dimensionless measure for characteristics
Standardized_moment
Particular case of the generalized extreme value distribution
is π / 6 ≈ 1.2825. {\displaystyle \pi /{\sqrt {6}}\approx 1.2825.} The cumulants, for n > 1, are given by κ n = ( n − 1 ) ! ζ ( n ) . {\displaystyle \kappa
Gumbel_distribution
Generalized chain rule in calculus
exponential Bell polynomial. In case g ( x ) {\displaystyle g(x)} is a cumulant-generating function, then f ( g ( x ) ) {\displaystyle f(g(x))} is a moment-generating
Faà_di_Bruno's_formula
Equation in statistical mechanics
{\displaystyle E[e^{-\beta W}]=e^{-\beta \Delta F}} , and use the cumulant expansion up to the second cumulant, we obtain E [ W ] − Δ F ≈ 1 2 β σ W 2 {\displaystyle
Jarzynski_equality
Family of continuous probability distributions
location-scale family can be made to fit the observed mean (first cumulant) and variance (second cumulant) arbitrarily well. However, it was not known how to construct
Pearson_distribution
Monte Carlo distribution shifting technique
M X ( θ ) {\displaystyle \kappa (\theta )=\log M_{X}(\theta )} be the cumulant-generating function (CGF). The exponentially tilted measure P θ {\displaystyle
Exponential_tilting
Fundamental result in the theory of large deviations
Cramér in 1938. The logarithmic moment generating function (which is the cumulant-generating function) of a random variable is defined as: Λ ( t ) = log
Cramér's theorem (large deviations)
Cramér's_theorem_(large_deviations)
Inequality in probability theory
Proof By the definition of variance proxy, it suffices to show that its cumulant generating function K ( t ) := log E [ e t ( X − E [ X ] ) ] {\displaystyle
Hoeffding's_lemma
Physical quantity
cumulant and thus it possesses the following properties: Cumulants can be explicitly represented only by moments of lower or equal order. Cumulants are
Total_position_spread
Empirical law on the variance of species in a habitat
distributed in accordance with a Poisson distribution. In the additive form its cumulant generating function (CGF) is: K b ∗ ( s ; θ , λ ) = λ κ b ( θ ) [ ( 1 +
Taylor's_law
(i + 1)-st moment. To use moment closure, a level is chosen past which all cumulants are set to zero. This leaves a resulting closed system of equations which
Moment_closure
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